How can I align two faces in edit mode
I think the answer is: yes you can.


Select the connecting edge and snap the 3D cursor to it
Set the pivot point to 3D Cursor
The connecting edge still selected, add a custom transformation orientation and keep it as orientation


Now


Select the first triangle (the target)
Use ShiftNumpad 1 or ShiftNumpad 3 to be in orthographic view aligned with this triangle and so that the connecting edge is aligned to the view. This is useful in order to snap as we want later.


To snap:


Enable snapping, set it to vertex, snap with closest, project onto self and enable it for rotation


To rotate:


Select the opposite vertex (the one of the other triangle)
Rotate R, hit Y to rotate around the connecting edge
Set the mouse cursor onto the opposite vertex of the target triangle


To rotate to the opposite:


Rotate R, hit Y, then hit 180 on the numpad